The structure of the lattice of normal extensions of modal logics with cyclic axioms

Irreflexive frames sometimes play a crucial role in the theory of modal logics, although the class of all such frames that consist of only irreflexive points can not be determined by any set of modal formulas. For instance, the modal logic determined by the frame of one irreflexive point is one of the two coatoms of the lattice of all normal modal logics. Another important result is that every rooted cycle-free frame, that consists of irreflexive points only, splits the lattice of all normal modal logics. In this paper, we consider a family of axioms Cycl(n) (for n 0), which forces frames to be n-cyclic. Seeking out the distribution of modal logics of irreflexive frames in the lattice of normal extensions of the modal logic with a cyclic axiom gives us information about the structure of this lattice. We mainly discuss the case n = 1 (the structure of the lattice of normal extensions of K Cycl(1)) and the case n = 2 (that of normal extensions of K Cycl(2)). Finally we discuss the possibility that a similar or a refined argument may bring us information on the structure of the lattice of normal extensions of the logic K Cycl(n) for every n 1.